In the aggregate meta-analysis of randomized clinical trials, one may attempt to assess treatment effects in a general population using the aggregate information (means, variances, proportions, etc.) only provided in each study. If the parameter of interest is the average causal effect of a treatment on an outcome, analysis using aggregate data is sufficient. However, when the parameters of interest are those of a marginal structural model (MSM) describing the treatment effect modification caused by another covariate, the parameters of the aggregate level model may be different from those of the individual level model. If so, the estimates from the aggregate data are biased for the true effect modification in the MSM. This is called "ecological bias".
The natural solution to eliminate this bias would be to base the meta-analysis not on the aggregate data, but rather on the raw data from each study. Unfortunately, investigators conducting a meta-analysis rarely have access to raw data from all studies. Estimating the MSM using only studies with raw data available is feasible, but this approach is prone to selection bias. We therefore propose to circumvent the problem by adjusting the estimators obtained at the aggregate level. This is done using information describing the relationship between the outcome and the effect modifier that can be obtained from the individual patient data.
In this thesis, we describe the source of ecological bias in the estimation of a treatment-specific MSM for a continuous outcome using the counterfactual perspective of causal inference. We propose estimators for the marginal regression parameters that are adjusted for ecological bias using the individual level regression parameters of a subsample of studies. We prove the convergence of these adjusted estimators. The effectiveness of the bias correction as well as the extent of the ecological bias in several scenarios are evaluated numerically through a simulation study.